Change of Variables Bound on Integral

Question

Let $f \in L^p_{\mathbb{P}}(\mathbb{R}^n)$ and let $u:\mathbb{R}^n\to \mathbb{R}^n$ be continuous. Can we bound $$ \int_{x \in \mathbb{R}^n} |f\circ u(x)|^p dx \leq C \int_{x \in \mathbb{R}^n} |f(x)|^p dx, $$ for some universal constant C depending only on $u$, $n$, and $p$? If not in general, then in what cases where $u(x) \neq k x$ (for some $k \in \mathbb{R}$).